Intertemporal Substitution in Macroeconomics:
Evidence from a Two-Dimensional Labor Supply
Model with Money∗
Ali Dib†
Louis Phaneuf‡
December 15, 2005
Abstract
The representative agent model of aggregate labor supply, and its cornerstone,
the hypothesis of intertemporal substitution in labor supply, have a history of
empirical failure when confronted with aggregate time-series data. We show that
a model in which the representative agent has preferences defined in terms of
consumption, leisure in the workweeks, leisure in non working weeks and real
balances overcomes most difficulties previously encountered by empirical studies
of intertemporal Euler equations. The overidentifying restrictions implied by
the model are far from rejected. The estimated parameters of preferences are
generally stable and meaningful. Furthermore, the estimated wage elasticities of
labor supply are much higher than previously found in the literature.
JEL classification: E13, E24, E32, J22
Keywords: Aggregate labor supply; Intensive margin; Extensive margin; Intertemporal
substitution; Real balances.
∗
Phaneuf acknowledges the financial support of FQRSC and SSHRC. The views expressed in this
paper are ours. No responsibility for them should be attributed to the Bank of Canada.
†
International Department, Bank of Canada. 234 Wellington St. Ottawa, ON. K1A 0G9, Canada.
Phone: (613) 782 7851. Email: ADib@bankofcanada.ca
‡
Department of Economics, Université du Québec à Montréal, P.O. Box 8888, Station Downtown, Montreal, Canada, H3C 3P8, and CIRPÉE. Phone: (514) 987 3000 ext. 3015. Email: phaneuf.louis@uqam.ca
1.
Introduction
Following the seminal work of Lucas and Rapping (1969), a large class of macroeconomic models have relied on the hypothesis of intertemporal elasticity of substitution
to explain fluctuations in aggregate employment in postwar industrialized economies.1
This hypothesis, which is the cornerstone of representative agent models of aggregate
labor supply, claims that the cyclical variations in employment result from the optimal
decisions of a representative household that substitutes hours worked intertemporally
in response to transitory movements in wage and interest rates. Thus, cyclical employment fluctuations are modelled as movements along a labor supply curve. Paradoxically, studies that use aggregate data to test the intertemporal-substitution-in-laborsupply (ISLS) hypothesis, find, in general, no supportive evidence, and often reach
negative conclusions.2 The evidence in Mankiw, Rotemberg, and Summers (1985), for
example, is typical of the problems encountered with the representative agent model of
aggregate supply over the years: the overidentifying restrictions implied by the theory
are almost always rejected, the estimated parameters of preferences are highly unstable, and the utility function is often not concave, leading to elasticities of the wrong
sign [see also Hall (1980), Altonji (1982), Killingsworth and Heckman (1986), Pencavel
(1986), and Eichenbaum, Hansen and Singleton (1988)].
Alogoskoufis (1987) argues that one possible explanation for the apparent lack of
success of representative agent models of aggregate labor supply is that most studies
have measured movements in labor supply solely by variations in weekly hours or
1
This includes real business cycle models, dynamic general equilibrium (DGE) models with sticky
nominal prices, models of labor market search, and limited-participation models, among others. In
these theoretical models, fluctuations in aggregate employment are modelled as movements along a
labor supply curve.
2
Card (1994) concludes that microeconomic evidence does not support the ISLS hypothesis either.
1
intensive margin. Yet, fluctuations in employee hours should also reflect changes in the
number of employees or extensive margin. Indeed, estimating an equation for log-linear
labor supply in which work effort is measured by the number of workers, rather than
by the hours worked per person, Alogoskoufis (1987) obtains findings which lend more
credence to the ISLS hypothesis. However, no attempt is made to formally incorporate
the optimal choices of employment at both margins into a single coherent framework
or to test the overall plausibility of the model per se.
Our paper proposes a tractable framework which allows a formal test of the ISLS
hypothesis in the context of a representative agent framework that explicitly accounts
for the adjustment of employment at the intensive and extensive margins. Further,
it explores the intertemporal labor-supply relationships implied by monetary economy
models, in line with several recent microfounded models which account for money,
by assuming that the representative agent derives utility from holding real balances.
Based on U.S. postwar data, we show that a model with these features overcomes most
problems previously encountered by empirical studies of intertemporal Euler equations.
Our model borrows some inspiration from the RBC literature, especially from the
work of Bils and Cho (1994) and Cho and Cooley (1994). These authors develop
RBC models which permit adjustment along both labor supply margins by realistically
making hours per week and weeks per year of leisure imperfect substitutes. This feature
is achieved by assuming that agents, who otherwise have identical preferences and
opportunity sets, have a fixed cost associated with labor supply that depends on the
fraction of days in a period that they will be employed.3 While our setting is broadly
consistent with theirs, our main objective is not to study the business cycle properties
of a calibrated version of the model as they do, but rather, to estimate the parameters
3
Cho and Rogerson (1988) also allow adjustment at both margins by introducing heterogeneity in
the opportunity sets of household decision makers.
2
of preferences of the two-dimensional aggregate labor supply model and to assess the
overall plausibility of the model.
The intertemporal Euler equations implied by our model, and some alternative models discussed below, are estimated with the Generalized Method of Moments. While
estimating the intertemporal Euler equations, we assume that the representative household can hold different types of riskless and/or risky assets: namely, bonds, shares, or
both. We also look at the sensitivity of our findings to different wage rates. Our main
findings can be summarized as follows.
First, we find that the distinction between leisure in the workweeks and leisure
in non-working weeks is strongly supported by aggregate data; each type of leisure
contributes substantially to the representative household’s utility according to their
estimated shares in the utility function. The inclusion of real balances in the utility function also finds empirical support. Importantly, the overidentifying restrictions
implied by the two-dimensional ISLS models with money easily survive formal, statistical, tests. These findings are not too sensitive to specific measures of asset returns
and wages.
Also, according to our estimates, the labor supply elasticity with respect to transitory movements in wages is higher for the weekly hours than for the weeks worked.
Combining the two elasticities, our model delivers a wage elasticity of labor supply
for total hours worked that, at the lowest, is 1.54 and can be as high as 2.15. In
comparison, the highest wage elasticity of labor supply previously found with aggregate time-series data was around unity [see Alogoskoufis (1987) and Cho, Merrigan
and Phaneuf (1998)]. We also find that asset returns have a significant independent
influence on total hours worked.
In an effort to better identify the key factors to our findings, we compare the results
3
of the two-dimensional ISLS models with real balances with those of alternative ISLS
models, including one-dimensional ISLS models with and without money, and twodimensional ISLS models without money. ISLS models that feature only one type of
leisure, whether they include real balances or not, do not perform well; their problems
range from non-concavity of the estimated utility functions to systematic rejection of
the overidentifying restrictions implied by these models. The two-dimensional ISLS
models that exclude real balances are also systematically rejected, even though they
perform better than one-dimensional ISLS models.
In related work, Rogerson and Rupert (1991) use microeconomic data from the
PSID for married males between the ages of 25 and 65 for the years 1976–82 to assess
the household’s willingness to substitute work intertemporally. In their model, workers
also choose weeks of work and hours of work per week. Focusing on the choice of
weeks, they show that allowing some individuals to be at a corner solution for weeks
worked, as a result of working year round, may substantially increase the intertemporal substitution elasticity of labor supply. Unlike Rogerson and Rupert (1991), we
employ aggregate time series rather than microeconomic data, provide estimates of the
preferences parameters, formally test the overidentifying restrictions implied by the
theoretical models, and do not focus on individuals who are at a corner solution for
weeks worked. Still, our evidence is consistent with theirs, suggesting that the representative agent model of labor supply is more plausible when labor supply decisions
are made two-dimensional.
The rest of this paper is organized as follows.
Section 2 constructs the two-
dimensional ISLS model with real balances and the alternative ISLS models. Section
3 describes the econometric methods and data used to estimate the models. Section
4 reports the empirical results of the two-dimensional ISLS model with real balances
4
and compares them with those of alternative ISLS models. Section 5 reports the intertemporal elasticities implied by the two-dimensional labor supply model with money.
Section 6 offers concluding remarks.
2.
The Models
2.1
A two-dimensional aggregate labor supply model with real
balances
Actual movements in employment can be measured by the sum of changes in labor
force participation and in the fraction of weeks that workers in the labor force are at
work. Following Bils and Cho (1994) and Cho and Cooley (1994), we focus on the
latter component. Hence, we do not consider how workers and nonworkers differ in
compensation and consumption in a competitive equilibrium. Further, some evidence
suggests that movements in employment rates for a given labor force are cyclically
more important than movements in labor force participation. For instance, Bils and
Cho (1994) report postwar U.S. evidence indicating that a 1 percent deviation from
trend in employment has resulted, on average, from a 0.8 percent change in the weeks
at work during the year for a given sized workforce and from a 0.2 percent change in
the fraction of individuals who worked during the year.4 Also, using microeconomic
data from the Michigan Panel Study of Income Dynamics (PSID), they show that the
distinction between leisure time in workweeks and weeks off in the year seems to be
empirically supported.
Hence, while our model does not seek to establish a distinction between people who
are in or out of the labor force, it assumes that weekly hours worked and weeks worked
4
In a similar vein, Hall and Lilien (1986) have shown that changes in the unemployment rate
have contributed 70 percent of the variation in total employment, whereas changes in labor force
participation have contributed 30 percent.
5
are imperfect substitutes. Specifically, let us consider an economy inhabited by a large
number of identical households whose preferences are defined over expected streams
of consumption, ct , leisure in the workweeks, l1t , leisure in the non-working weeks, l2t ,
and real balances, mt . The representative household maximizes the following lifetime
expected discounted utility:
Et
∞
X
β t u (ct , l1t , l2t , mt ) ,
(1)
t=0
where uc , ul1 , ul2 , um > 0 and ucc, ul1 l1 , ul2 l2 , umm < 0, β is the rate of time discount and
Et denotes the mathematical expectation.
In this two-dimensional aggregate labor supply framework, money holding generates
a positive utility. One possible interpretation is that money facilitates consumption and
reduces the time allocated for shopping.5 Further, as we will see below, the presence of
money in the utility function also implies an extra dynamic Euler equation associated
with the optimal choice of real balances that can be used in estimating the model.
In each period, the representative household’s total time endowment is a product
of the number of weeks, E, and the hours available in the week, H, with both E and H
taken as given. The representative household allocates its time during the workweeks,
et , to work ht (the weekly hours per worker), leisure l1t , and a fixed cost associated with
labor supply τ . In non-working weeks, time is entirely devoted to leisure l2t . Hence,
leisure in the workweeks is
l1t = (H − ht − τ ) et ,
(2)
while leisure in non-working weeks is given by
l2t = (E − et ) H.
5
(3)
Feenstra (1986) establishes a functional equivalence between liquidity costs and the utility of
money.
6
The representative household ranks alternative streams of consumption, leisure in
the workweeks, leisure in non-working weeks, and real balances, using the following
constant relative risk aversion (CRRA) utility function:
"
#
∞
αc αl1 αl2 αm 1−σ
X
(c
l
l
m
)
−
1
t 1t 2t
t
βt
,
Et
1
−
σ
t=0
(4)
where αc + αl1 +αl2 + αm = 1. Concavity requires that αc , αl1 , αl2 , and αm have positive
signs, and that the product of each of these exponents with (1 − σ) be less than unity.6
The representative consumer’s allocations must satisfy the following sequence of
budget constraints:
pt ct + pt mt + bt + st ≤ wt ht et + rbt bt−1 + rst st−1 + pt−1 mt−1 + trt ,
(5)
where pt is the price of the consumption good in period t, bt and st are bonds and
shares (riskless and risky assets, respectively) purchased in period t, wt is the after-tax
wage rate, rbt is the after-tax riskless asset return, rst =
pst +dst
pst−1
is the after-tax risky
asset return, pst is the price of shares, dst is the dividends earned by shares in period t,
and trt is a lump-sum money transfer. The budget constraint (5) describes a situation
where the representative consumer holds bonds and shares. However, in deriving the
intertemporal Euler equations below, we consider that the representative consumer can
possibly hold bonds, shares, or both.
The representative household chooses ct , ht , et , mt , bt , and st that maximize expected
total discounted utility (4) subject to (5), while taking into account (2) and (3). The
6
Eichenbaum, Hansen, and Singleton (1988) describe in more detail the desirable characteristics
of the above form of utility function.
7
first-order conditions are:
αl1 αl2 αm 1−σ −1
αc (cαt c l1t
l2t mt )
ct
− λt pt = 0,
(6)
αl1 αl2 αm 1−σ −1
αl1 (cαt c l1t
l2t mt )
l1t − λt wt = 0,
(7)
τ
) = 0,
H
− λt + βEt λt+1 = 0,
αl1 αl2 αm 1−σ −1
l2t mt )
l2t − λt wt (1 −
αl2 (cαt c l1t
αl1 αl2 αm 1−σ
αm (cαt c l1t
l2t mt )
(mt pt )−1
(8)
(9)
βEt λt+1 rbt+1 − λt = 0,
(10)
βEt λt+1 rst+1 − λt = 0,
(11)
where λt is the multiplier associated with the intertemporal budget constraint, which
also represents the marginal utility of wealth at date t.7
Assuming that λt follows a martingale process (see MaCurdy 1985), we derive the
following intertemporal Euler equations:
( ·µ
¶α µ
¶α µ
¶α µ
¶α ¸1−σ
ct+1 c mt+1 m l1t+1 l1 l2t+1 l2
Et β
ct
mt
l1t
l2t
)
¶−1
µ
pt rjt+1
ct+1
×
−1
= 0,
ct
pt+1
( ·µ
¶α µ
¶α µ
¶α µ
¶α ¸1−σ
ct+1 c mt+1 m l1t+1 l1 l2t+1 l2
Et β
ct
mt
l1t
l2t
)
¶−1
µ
l1t+1
wt rjt+1
×
−1
= 0,
l1t
wt+1
( ·µ
¶α µ
¶α µ
¶α µ
¶α ¸1−σ
ct+1 c mt+1 m l1t+1 l1 l2t+1 l2
Et β
ct
mt
l1t
l2t
)
¶−1
µ
l2t+1
wt rjt+1
×
−1
= 0,
l2t
wt+1
7
(12)
(13)
(14)
From equation (9), we can derive the standard money-demand function in which real balances
depend on consumption and interest rates.
8
( ·µ
¶α µ
¶α µ
¶α µ
¶α ¸1−σ
ct+1 c mt+1 m l1t+1 l1 l2t+1 l2
Et β
ct
mt
l1t
l2t
)
¶−1
µ
ct+1
αm ct
pt
×
+
−1
= 0,
ct
pt+1 αc mt
(15)
with rjt+1 , j = b, s or both, denoting the assets the representative agent holds. With
j = b, the representative consumer holds only riskless assets, and hence, there are four
Euler equations that can be estimated jointly to retrieve the preference parameters of
the model. The same applies if j = s, so that the representative consumer holds only
risky assets. With j = b and s, the representative consumer holds riskless and risky
assets, which implies seven intertemporal Euler equations.
Conditions (12)–(14) state that the marginal cost of one unit of consumption or of
an hour of both types of leisure in period t must equal the expected marginal benefit in
period t + 1. Condition (15), which follows directly from the inclusion of real balances
in the utility function, implies that the representative household compares its marginal
utility of holding an additional dollar in period t with the marginal disutility of not
consuming this dollar in the current period versus the expected discounted marginal
utility of consumption in the future. Note also that real balances appear in each
intertemporal Euler equation.
2.2
Alternative models
Our empirical investigation also entails a comparison of the two-dimensional aggregate labor supply model with money with some alternative models, namely, a standard model that features only one type of leisure and abstracts from money, the onedimensional aggregate labor supply model with money, and the two-dimensional labor
supply model without real balances. For simplicity, we provide only the intertemporal
Euler equations associated with these models.
9
2.2.1 The standard aggregate labor supply model (no money)
The standard aggregate labor supply model does not rely on the distinction between
leisure in the workweeks and leisure in non-working weeks. It uses instead the standard
measure of leisure and does not account for real balances. In the standard model, the
representative household’s preferences are described by,
"
#
∞
αc αl 1−σ
X
(ct lt )
−1
,
βt
Et
1−σ
t=0
(16)
where αc + αl = 1. Leisure is defined as lt = (E × H) − nt , with nt representing the
total hours worked during period t. The representative household faces the following
budget constraint:
pt ct + bt + st ≤ wt nt + rbt bt−1 + rst st−1 .
(17)
Optimal choices of ct , nt , bt , and st are those that maximize expected total discounted
utility (16) subject to (17). The intertemporal Euler equations corresponding to this
optimization problem are:
( ·µ
)
¶α µ
¶α ¸1−σ µ
¶−1
ct+1 c lt+1 l
ct+1
pt rjt+1
−1
= 0,
Et β
ct
lt
ct
pt+1
( ·µ
)
¶α µ
¶α ¸1−σ µ
¶−1
ct+1 c lt+1 l
lt+1
wt rjt+1
Et β
−1
= 0,
ct
lt
lt
wt+1
(18)
(19)
for rjt+1 , j = b, s or both. With j = b or s, the model yields two intertemporal Euler
equations, while with j = b and s, it delivers four.
2.2.2 The one-dimensional aggregate labor supply model with real balances
In the one-dimensional aggregate labor supply model with real balances, preferences
are given by
10
Et
∞
X
t=0
"
βt
#
(cαt c ltαl mαt m )1−σ − 1
,
1−σ
(20)
where αc + αl + αm = 1. The representative agent has the following budget constraint:
pt ct + pt mt + bt + st ≤ wt nt + rbt bt−1 + rst st−1 + pt−1 mt−1 + trt .
(21)
The representative agent’s optimization problem consists in choosing ct , nt , mt , bt , and
st that maximize (20) subject to (21). The corresponding intertemporal Euler equations
are:
( ·µ
)
¶α µ
¶α µ
¶α ¸1−σ µ
¶−1
ct+1 c lt+1 l mt+1 m
ct+1
pt rjt+1
−1
= 0,(22)
Et β
ct
lt
mt
ct
pt+1
( ·µ
)
¶α µ
¶α µ
¶α ¸1−σ µ
¶−1
ct+1 c lt+1 l mt+1 m
lt+1
wt rjt+1
Et β
−1
= 0,(23)
ct
lt
mt
lt
wt+1
( ·µ
)
¶α µ
¶α µ
¶α ¸1−σ µ
¶−1
ct+1 c lt+1 l mt+1 m
ct+1
αm ct
pt
Et β
+
−1
= 0,(24)
ct
lt
mt
ct
pt+1 αc mt
for rjt+1 , j = b, s or both. Hence, the one-dimensional labor supply model with real
balances yields three Euler equations with j = b or s, and five Euler equations with
j = b and s.
2.2.3 The two-dimensional aggregate labor supply model without real balances
The third model incorporates both types of leisure, l1t and l2t , while omitting real
balances. The representative agent’s preferences are described by,
"
#
∞
αc αl1 αl2 1−σ
X
−1
t (ct l1t l2t )
,
Et
β
1−σ
t=0
(25)
where αc + αl1 + αl2 = 1. The budget constraint is,
pt ct + bt + st ≤ wt ht et + rbt bt−1 + rst st−1 .
11
(26)
The representative household chooses ct , ht , et , bt , and st that maximize expected
discounted utility (25) subject to (26), while taking into account definitions (2) and
(3). The intertemporal Euler equations corresponding to this optimization problem
are:
( ·µ
)
¶ αc µ
¶αl1 µ
¶αl2 ¸1−σ µ
¶−1
l1t+1
l2t+1
ct+1
ct+1
pt rjt+1
−1
= 0, (27)
Et β
ct
l1t
l2t
ct
pt+1
( ·µ
)
¶ αc µ
¶αl1 µ
¶αl2 ¸1−σ µ
¶−1
l1t+1
l2t+1
l1t+1
ct+1
wt rjt+1
Et β
−1
= 0, (28)
ct
l1t
l2t
l1t
wt+1
( ·µ
)
¶ αc µ
¶αl1 µ
¶αl2 ¸1−σ µ
¶−1
l1t+1
l2t+1
l2t+1
ct+1
wt rjt+1
Et β
−1
= 0, (29)
ct
l1t
l2t
l2t
wt+1
for rjt+1 , j = b, s or both. This model delivers three Euler equations with j = b or s,
and six with j = b and s.
While, in principle, the intertemporal Euler equations can be estimated either individually or jointly, we focus on the joint estimation, given the large number of models
considered.
3.
Estimation Procedure and Data
This section describes the econometric procedure and data used to estimate the Euler
equations of our ISLS models.
3.1
Estimation method
The parameters of preferences of the alternative aggregate labor supply models are
retrieved from the intertemporal Euler equations which are estimated using the generalized method of moments (GMM) procedure proposed by Hansen (1982) and Hansen
and Singleton (1982). Let Et [q(yt+1 , θ)] be a vector of n Euler equations derived from
12
a particular model. The vector yt+1 is composed of stationary variables dated t and
t + 1, while the vector θ is composed of the l structural parameters that we seek to
estimate. For a vector zt of k instrumental variables included in the information set
available in period t, we define a vector of (n × k) unconditional moment restrictions
implied by the model,
Ef (yt+1 , zt , θ) = E[q(yt+1 , θ) ⊗ zt ],
(30)
where ⊗ is the Kronecker product. The sampling equivalence of equation (30) is given
by,
T
1X
gT (θ) =
f (yt+1 , zt , θ),
T t=1
(31)
with gT (θ) converging asymptotically towards zero under the null hypothesis that the
structural model is well specified. The GMM estimator of the parameter vector (θT )
is the solution to the following problem:
θT = arg min gT (θ)0 WT gT (θ),
θ
(32)
where WT is a non-negative symmetric weighting matrix. An optimal weighting matrix, WT∗ , is obtained as the inverse of the variance-covariance matrix of the moment
conditions evaluated at a consistent first-step estimator; for example, by using the identity matrix as the weighting matrix at the first step. This optimal weighting matrix is
consistently estimated with the procedure developed by Newey and West (1994).
The overidentifying restrictions implied by the different models can be tested formally using Hansen’s J-statistic if the dimension of the vector of moments is greater
than the dimension of the vector of estimated parameters. This statistic is given by:
J = T {gT (θ)0 WT∗ gT (θ)} ,
13
(33)
and asymptotically follows a χ2 distribution with nk − l degrees of freedom.
The list of instruments used in the estimation of aggregate labor supply models has
not been uniform across past studies. For example, Mankiw, Rotemberg, and Summers
(1985) use different combinations, including lagged consumption, lagged interest rates,
lagged leisure, lagged prices, and lagged wages.8 Eichenbaum, Hansen, and Singleton
(1988) employ the current rates of change of consumption, leisure, and wages, and the
current interest rate.
Following Tauchen (1986) and Kocherlakota (1990), we use only current and oneperiod lagged values of the instrumental variables to estimate the intertemporal Euler
equations. Unlike the models that have been tested by Mankiw, Rotemberg, and Summers (1985) and by Eichenbaum, Hansen, and Singleton (1988), our model includes two
different measures of leisure, money and different measures of asset returns. Hence, we
r
use different subsets of the following instrumental variables: {1, RctjtRpt ,
1
,
Rct Rpt
Rct ,Rl1t ,
Rl2t , Rlt , Rmt } (where Rxt = xt /xt−1 ).9
3.2
Data
The models are estimated with the help of quarterly (seasonally adjusted) data for
the period 1960Q1–1993Q4.10 Aggregate real per capita consumption, ct , is the sum
of consumption expenditures on non-durable goods and services, converted into per
capita terms after dividing by the total adult population (age sixteen and over). The
aggregate price level, pt , is the implicit price deflator corresponding to our measure of
consumption expenditures. The rate of return on riskless assets, rbt , is the 3-month
8
They occasionally use up to five-period lagged values of the variables.
We use Rl1t and Rl2t when estimating two-dimensional ISLS models, and Rlt in the estimation
of one-dimensional ISLS models.
10
Our choice of sample period is justified by the fact that we want to use hours worked from the
Household Survey to be consistent with previous studies. This series is unfortunately unavailable after
1993Q4.
9
14
Treasury bill rate, expressed in real terms. The rate of return on risky assests, rst , is
the value-weighted average of returns on the New York Stock Exchange, also expressed
in real terms. The monetary aggregate used to calculate real money balances is M1.
We use two different wage measures. The first one, represented by w1 , is the
average hourly compensation in non-agricultural employment. The second, represented
by w2 , adheres more closely to the National Income and Product Accounts (NIPA)
and is the sum of NIPA definitions of wages and salaries, other labor income, and
proprietary income divided by total labor hours following the suggestions in Dutkowsky
and Dunsky (1996).11 The wage rates and asset returns are both after-tax measures.
The representative household’s average tax rate is based upon the taxability properties
of the various components of disposable income.12
The representative household’s quarterly time endowment is 1,456 hours (13 weeks
× 112 hours).13 The weekly average hours worked, ht , is the hours series from the
Household Survey. As in Alogoskoufis (1987), we approximate the working weeks, et ,
by the product of the number of weeks in the quarter and the ratio of the civilian
employment to the working population. The working-time cost is set at 6 hours per
workweek.14
11
Thus, proprietary income is included entirely within labour compensation, although part of this
income could represent the return on capital.
12
Specifically, the average tax rate, ωt , comes from the following disposable income equation:
Y Dt = (1 − ωt )(W St + IN Tt ) + OLYt + T Pt ,
where Y D is nominal disposable income; W S is the sum of nominal wages and salaries and proprietary
income; IN T is the sum of total interest, dividends, and rental income less interest paid on household
debt; OLY denotes other labor income; and T P represents the nominal transfer payments. Other
labor income (which consists primarily of labor benefits) and transfer payments are tax exempt, to
be consistent with IRS tax laws. Solving the equation above for (1 − ωt ) gives the average tax rate
(see Dutkowsky and Dunsky 1996).
13
We adjust for sleeping time, so the representative household’s daily time endowment is 16 hours.
14
We have considered a range of 2 to 12 hours for the fixed time cost without any significant impact
on the results.
15
When estimating one-dimensional ISLS models, we measure total hours worked
during the quarter nt by (hourst × empt × 13) /popt , with hourst representing the average weekly hours, empt the weekly employees (total employed labour force), and popt
the total adult population.
4.
Results
We begin by reporting the estimation results obtained from the two-dimensional aggregate labor supply model with real balances. We then compare these results with
those of alternative models. Estimates of the parameters of preferences are recovered
from the jointly estimated intertemporal Euler equations. Tables 1 and 2 report the
results.
4.1
The two-dimensional aggregate labor supply model with
real balances
The preference parameters of two-dimensional ISLS models with money are retrieved
from the estimated intertemporal Euler equations (12)–(15). Table 1 reports the estimates. The instrumental variables used in estimating the models are listed at the
bottom of the table. Each of the three panels in this table reports findings that correspond to a particular assumption about the holding of assets. Also listed within each
panel are the results obtained with the two wage measures.
The overidentifying restrictions implied by the two-dimensional aggregate labor supply models with real balances are generally not rejected at a conventional confidence
level. The estimates of β, αc , αl1 , αl2 , αm , and σ always imply that the concavity conditions are satisfied. Note that the estimates are highly stable wirh different measures
of asset returns and wages. In contrast, Mankiw, Rotemberg, and Summers (1985) find
16
that the overidentifying restrictions implied by standard one-dimensional aggregate labor supply models are systematically rejected and that the estimated parameters of
preferences are highly unstable.
Panel A of Table 1 reports the estimated structural parameters with the representative consumer holding only risky assets, st . The estimates of the discount factor, β, are
below unity, a finding that is interesting in itself, since most previous empirical studies
of intertemporal Euler equations have found estimates of β that are above unity.
Our estimates confirm that the two dimensions of leisure are statistically significant,
each type of leisure contributing quite substantially to the household’s preferences, as
shown by their estimated shares in preferences. The estimated share of leisure in
working weeks is 0.34 when the wage rate is measured by w1 , and 0.36 when it is w2 .
The share of leisure in non-working weeks is 0.33 with both w1 and w2 . The estimates
of αc are 0.33 and 0.31, respectively. The estimated share of real balances, while small
at 0.002, is nonetheless statistically significant. The estimated preference parameter,
σ, which is also statistically significant, is 3.88 with wage rate w1 , and 5.89 with w2 .
Panel B of Table 1 reports the results obtained with the assumption that the representative household holds only riskless assets, bt . While the results are not as good as
those with the risky assets, they are still encouraging. The overidentifying restrictions
are not statistically rejected at a conventional confidence level, but their non-rejection
is not as strong as previously. The discount factor, β, is now slightly above unity, as
in most empirical studies of intertemporal Euler equations.
With rbt , the wage measure seems to have a greater impact on the estimated preference shares of consumption and leisure in working weeks. Estimates of αc are 0.44
with w1 , and 0.31 with w2 . The estimates of αl1 are 0.22 and 0.35, respectively. In
comparison, the estimated share of leisure in non-working weeks, αl2 , is stable when
17
we change the wage rate, with estimates of 0.34 and 0.33, respectively. The preference
share of real balances is 0.0018 or 0.0012, depending on the wage measure, and is statistically significant. The estimates of σ are broadly similar to those obtained with the
risky assets, at 3.90 with wage rate w1 and 5.94 with w2 .
Panel C of Table 1 reports the results with the representative household holding
riskless and risky assets. By far, the overidentifying restrictions implied by the model
are not refuted. The estimates of αc and αl1 are quite stable when the wage rate is
changed from w1 to w2 , with 0.46 and 0.39 for αc , and 0.22 and 0.29 for αl1 . Estimates
of αl2 are virtually unchanged at 0.32. The estimated share of real balances, αm , is
also stable at 0.0013 and 0.0016. Estimates of σ are 4.04 with w1 and 5.45 with w2 .15
The findings reported in this subsection suggest that allowing the representative
consumer to have preferences defined in terms of consumption, leisure in the workweeks,
leisure in non-working weeks, and real balances results in a better overall fit of U.S.
aggregate time-series data by the representative agent model of aggregate labor supply.
In particular, the overidentifying restrictions implied by the model cannot be rejected
and the estimated parameters of preferences are highly stable even when we use different
measures of asset returns and wages.
4.2
Alternative aggregate labor supply models with and without money
It is also possible to assess indirectly the empirical success of the two-dimensional
aggregate labor supply model with money by estimating alternative versions of the
model that feature less theoretical ingredients than in the more general framework.
15
When β was found to be slightly above unity, the corresponding Euler equations were re-estimated,
constraining the parameter β to be below unity, with almost no changes in the estimates of the other
structural parameters of the model.
18
Table 2 reports the results of one-dimensional labor supply models, with and without
real balances, and of two-dimensional labor supply models without money.16
Panel A of Table 2 reports the estimates of one-dimensional labor supply models
that abstract from money (or standard models). The estimated intertemporal Euler
equations are (18) and (19). As in Mankiw, Rotemberg, and Summers (1985), Hansen’s
J-test strongly rejects the overidentifying restrictions implied by the standard models.
Assuming that the representative consumer holds riskless and risky assets, the estimate
of σ is −2.56, which violates the concavity conditions.
Panel B of Table 2 reports the results of one-dimensional labor supply models with
real balances. The estimated structural parameters are recovered from the Euler equations (22)–(24). The results are still very negative, with overidentifying restrictions
that are strongly rejected. The concavity conditions are not satisfied when the representative consumer holds only risky assets or a combination of riskless and risky assets,
the estimates of σ being −2.73 and −6.98, respectively. Note also that these estimates
are highly unstable. While the estimate of αc is 0.32 with the riskless assets, it drops
to 0.044 with the risky assets. Altogether, these negative results constitute strong
evidence against one-dimensional aggregate labor supply models.
Panel C of Table 2 reports estimates of two-dimensional ISLS models that abstract
from real balances, based on the intertemporal Euler equations (27)–(29). As before,
the overidentifying restrictions of the model are strongly rejected. Note also that the
omission of money from the two-dimensional aggregate labor supply model greatly
affects the estimates of σ, which now becomes unstable. Indeed, estimates of σ are
51.58 with the risky assets, 4.11 with the riskless assets, and 19.66 with riskless and
16
Since the results of one-dimensional ISLS models and two-dimensional ISLS models without money
are essentially negative, we report only those obtained with wage rate w1 . The alternative models have
been estimated using the same set of instruments as in the more general model or different subsets of
instruments with the same negative results.
19
risky assets held jointly.
The findings presented in this subsection thus provide indirect empirical support
to the claim that the two dimensions of the aggregate labor supply decisions and the
insertion of real balances in the utility function are key factors contributing to the
empirical success of the two-dimensional aggregate labor supply model with money.
5.
Intertemporal Elasticities
Using the estimates reported in Table 1 of the two-dimensional aggregate labor supply models with real balances, we can derive the short-run elasticities of consumption,
weekly hours worked, weeks worked, and real balances with respect to transitory movements in wage rates, asset returns, and prices. To compute these elasticities, we assume
that transitory changes in wages, asset returns, and prices have only one-period effects.
MaCurdy (1981) assumes that an unanticipated transitory change in the wage rate triggers a wealth effect. However, under the assumption that the wealth effect is negligible
and has no long-run effect, McLaughlin (1995) shows that the constant elasticity of
marginal wealth (λ-constant elasticity) is equivalent to the short-run, compensated
elasticity of substitution.17
Assuming a deterministic environment and a small wealth effect, the elasticities can
be derived as in Mankiw, Rotemberg, and Summers (1985) and McLaughlin (1995).18
The intertemporal Euler equations are first transformed into log-linear systems, and
are then differentiated with respect to the log of endogenous and exogenous variables.
17
With transitory shocks, infinite lifetime, and a low discount rate, which are features of our model,
the wealth effect is indeed negligible.
18
With this assumption, there is certainty equivalence in the Euler equations, which allows us to
ignore the expectation operator (Mankiw, Rotemberg, and Summers 1985; Alogoskoufis 1987).
20
This gives the following short-run, intertemporal substitution matrix:
µ
¶ µ
¶−1 µ
¶
∂ ln yt
∂ ln qt
∂ ln qt
=
,
∂ ln xt
∂ ln yt
∂ ln xt
where yt is a vector of endogenous variables such that yt = (ct , mt , ht , et )0 ; vector
xt = (wt , rjt+1 , pt )0 is composed of exogenous variables, and vector qt , of the Euler
equations.
Table 3 reports the intertemporal elasticities that are based on the estimates of the
two-dimensional aggregate labor supply models with money. The estimated short-run
elasticities are generally lower than unity, except for the weekly hours of work and the
total hours worked. The signs of the elasticities are what the theory predicts.
A transitory rise in wages generates an increase in consumption, weekly hours
worked, weeks worked, and real balances. The wage elasticity of weekly hours worked
is in the range of 1.45–1.72 with wage rate w1 , and 1.24–1.44 with w2 . The wage
elasticity of working weeks is between 0.36 and 0.43 with w1 , and 0.31 and 0.36 with
w2 .
Combining the two elasticities for the weekly hours worked and the working weeks,
the wage elasticity of total hours worked falls in the range of 1.81–2.15 with w1 , and
1.54–1.81 with w2 . These elasticities are much higher than those previously reported
in the literature using aggregate time-series data.
A transitory increase in asset returns raises labor supply, while reducing both consumption expenditures and real money balances. The asset-return elasticity of total
hours worked lies between 0.48 and 0.63 with w1 , and between 0.27 and 0.29 with w2 .
These elasticities are consistent with those obtained by Alogoskoufis (1987).
21
6.
Conclusion
The representative agent model of aggregate labor supply, and its cornerstone, the
hypothesis of intertemporal substitution in labor supply, have not been very successful
when confronted with aggregate time-series data. The findings reported here suggest
that some credibility in this general framework may be restored if the representative
agent has preferences defined in terms of consumption, leisure in workweeks, leisure in
non working weeks and real balances.
Future work should allow for non-separability of preferences that would accommodate either intertemporal substitution or complementarity of leisure using a twodimensional aggregate labor supply framework of the kind we have proposed in this
paper.
22
References
Alogoskoufis, G., 1987, “On Intertemporal Substitution and Aggregate Labor Supply.”
Journal of Political Economy 95: 938–60.
Altonji, J., 1982, “The Intertemporal Substitution Model of Labor Market Fluctuations: An Empirical Analysis.” Review of Economic Studies 49: 783-824.
Bils, M. and J.O. Cho, 1994, “Cyclical Factor Utilization.” Journal of Monetary Economics 33: 319–54.
Card, D., 1994, “Intertemporal Labor Supply: An Assessment.” In Advances in Econometrics, Sixth World Congress, edited by C. Sims. New York: Cambridge University Press.
Cho, J.O. and T.F. Cooley, 1994, “Employment and Hours Over the Business Cycle.”
Journal of Economic Dynamics and Control 18: 411-432.
Cho, J.-O., P. Merrigan, and L. Phaneuf, 1998, “Weekly Employee Hours, Weeks
Worked and Intertemporal Substitution.” Journal of Monetary Economics 41: 185–
99.
Cho, J.O. and R. Rogerson, 1988, “Familiy Labor Supply and Aggregate Fluctuations.”
Journal of Monetary Economics 21: 233-245.
Dutkowsky, D.H. and R.M. Dunsky, 1996, “Intertemporal Substitution, Money and
Aggregate Labor Supply.” Journal of Money, Credit and Banking 28: 216–32.
Eichenbaum, M., L.P. Hansen, and K.J. Singleton, 1988, “A Time Series Analysis of
Representative Agent Models of Consumption and Leisure Choice under Uncertainty.” Quarterly Journal of Economics 103: 51–78.
Feenstra, R.C., 1986, “Functional Equivalence between Liquidity Costs and the Utility
of Money.” Journal of Monetary Econometrics 17: 271–91.
23
Hall, R.E., 1980, “Labor Supply and Aggregate Fluctuations.” Carnegie-Rochester
Conference Series of Public Policy 12: 7-33.
Hall, R.E. and D.M. Lilien, 1986, “Cyclical Fluctuations in the Labor Market.” In
The Handbook of Labor Economics, vol. 1, edited by O. Ashenfelter and R. Layard.
Amsterdam: North-Holland.
Hansen, L.P., 1982, “Sample Properties of Generalized Method of Moment Estimators.”
Econometrica 50: 1029–54.
Hansen, L.P. and K.J. Singleton, 1982, “Generalized Instrumental Variables Estimation
of Nonlinear Rational Expectations Models.” Econometrica 50: 1269–85.
Killingsworth, M. and J.J. Heckman, 1986, “Female Labor Supply: A Survey.” In The
Handbook of Labor Economics, vol. 1, edited by O. Ashenfelter and R. Layard.
Amsterdam: North-Holland.
Kocherlakota, N.R., 1990, “On Tests of Representative Consumer Asset Pricing Models.” Journal of Monetary Economics 26: 285–304.
Lucas, R.E., Jr. and L.A. Rapping, 1969, “Real Wages, Employment and Inflation.”
Journal of Political Economy 77: 721–54.
MaCurdy, T.E., 1981, “An Empirical Model of Labor Supply in a Life-Cycle Setting.”
Journal of Political Economy 89: 1059-85.
, 1985, “Interpreting Empirical Models of Labor Supply in an Intertemporal
Framework with Uncertainty.” In Longitudinal Analysis of Labor Market Data,
edited by J.J. Heckman and B. Singer, Econometric Society Monograph Series, vol.
10. Cambridge: Cambridge University Press.
Mankiw, N.G., J.J. Rotemberg, and L.H. Summers, 1985, “Intertemporal Substitution
in Macroeconomics.” Quarterly Journal of Economics 100: 225–51.
24
McLaughlin, K., 1995, “Intertemporal Substitution and λ-constant Comparative Statics.” Journal of Monetary Economics 35: 99–123.
Newey, W.K. and K.D. West, 1994, “Automatic Lag Selection in Covariance Matrix
Estimation.” Review of Economic Studies 61: 631–53.
Pencavel. J., 1986, “Labor Supply of Men: A Survey.” In The Handbook of Labor
Economics, vol. 1, edited by O. Ashenfelter and R. Layard. Amsterdam: NorthHolland.
Rogerson, R. and P. Rupert, 1991, “New Estimates of Intertemporal Substitution:
The Effect of Corner Solutions for Year-Round Workers.” Journal of Monetary
Economics 27: 255–69.
Tauchen, G., 1986, “Statistical Properties of Generalized Method-of-Moments Estimators of Structural Parameters Obtained from Financial Market Data.” Journal of
Business and Economic Statistics 4: 397-425.
25
Table 1:
Estimates of the Two-Dimensional ISLS Models with Money
β
αc
αl1
αl2
αm
σ
J-stat.
A. Euler equations with risky asset return (rst )
w1
0.9971
(0.005)
0.3308
(0.171)
0.3380
(0.138)
0.3289
(0.054)
0.0023
(0.0012)
3.8798
(1.778)
27.87
(0.22)
w2
0.9991
(0.005)
0.3100
(0.104)
0.3576
(0.085)
0.3304
(0.033)
0.0021
(0.0007)
5.8916
(1.916)
26.99
(0.25)
B. Euler equations with riskless asset return (rbt )
w1
1.0078
(0.002)
0.4427
(0.221)
0.2197
(0.196)
0.3358
(0.055)
0.0018
(0.0009)
3.8996
(1.807)
28.85
(0.068)
w2
1.0092
(0.002)
0.3149
(0.093)
0.3522
(0.079)
0.3316
(0.0320)
0.0012
(0.0003)
5.9384
(1.927)
26.50
(0.12)
C. Euler equations with both asset returns (rst and rbt )
w1
w2
1.0073
0.4651
0.2152
0.3176
0.0013
4.0440
38.14
(0.001)
(0.203)
(0.185)
(0.037)
(0.0004)
(1.560)
(0.247)
1.0086
(0.001)
0.3903
(0.110)
0.2922
(0.095)
0.3158
(0.028)
0.0016
(0.0005)
5.4495
(1.532)
35.55
(0.349)
Notes: Standard errors are in parentheses. P -values are in parentheses under J-statistics; the
wage rates w1 and w2 are defined in the text. The vectors of instrumental variables, zt , used
to estimate the models are: in A, zt = (1, Rct , Rmt , Rl1t , Rl2t , RcrtstRpt , Rct1Rpt )0 ; in B, zt =
(1, Rct , Rmt , Rl1t , Rl2t , RcrtstRpt )0 ; in C, zt =(1, Rct1Rpt , Rct , Rmt , Rl1t , Rl2t )0 .
26
Table 2:
Estimates of Alternative ISLS Models
β
αc
αl1
αl2
αm
αl
σ
J-stat.
A. Standard ISLS model
rs
rb
rs,b
rs
rb
rs,b
rs
rb
rs,b
1.0069
(0.030)
1.0129
(0.005)
0.9974
(0.005)
0.4143
0.5857
1.7980
(0.201)
(0.201) (19.26)
0.2272
0.7728
10.156
(0.053)
(0.054) (4.731)
0.2942
0.7058
-2.558
(0.139)
(0.140) (4.52)
B. One-dimensional ISLS model with money
27.28
(0.0001)
26.80
(0.0001)
28.46
(0.0008)
0.9913
(0.009)
1.0098
(0.0003)
0.9914
(0.005)
0.0438
0.0055
0.9517 -2.7282
(0.447)
(0.057) (0.478) (7.4111)
0.3227
0.0006
0.6772
6.8063
(0.070)
(0.0001) (0.070) (2.762)
0.2142
0.0005
0.7857 -6.9859
(0.060)
(0.0001) (0.060) (4.086)
C. Two-dimensional ISLS model (no money)
36.70
(0.0008)
33.10
(0.0005)
35.48
(0.0034)
1.0685
(0.022)
1.0057
(0.002)
1.0255
(0.003)
0.3718
(0.129)
0.2708
(0.150)
0.3154
(0.085)
25.61
(0.007)
25.35
(0.008)
30.63
(0.060)
0.3620
0.068
0.3367
(0.141)
0.3441
(0.073)
0.3200
(0.033)
0.3925
(0.052)
0.3405
(0.029)
-
-
-
-
-
-
51.580
(23.61)
4.1062
(2.049)
19.657
(7.948)
Notes: Standard errors are in parentheses. P -values are in parentheses under J-statistics; the wage
rate is w1 , defined in the text. The instrumental variables used to estimate the alternative models are
r
taken from the following set of variables: {1, RctjtRpt , Rct1Rpt , Rct , Rlt , Rl1t , Rl2t , Rmt }, j = b, s, and
Rxt = xt /xt−1 .
27
Table 3:
Elasticities Implied by the Two-Dimensional ISLS Models with Money
A. Risky return, rst
wt
rt+1
pt
B. Riskless return, rbt
wt
rt+1
pt
C. Both return rates
wt
rt+1
pt
ct
w1
w2
0.495
0.571
-0.176
-0.080
-0.753
-0.741
0.413
0.568
-0.148
-0.076
-0.669
-0.737
0.402
0.500
-0.132
-0.075
-0.649
-0.680
ht
w1
w2
1.454
1.235
0.508
0.231
-0.712
-0.746
1.690
1.242
0.428
0.220
-0.952
-0.757
1.724
1.444
0.382
0.215
-1.011
-0.922
et
w1
w2
0.360
0.305
0.126
0.057
-0.176
-0.184
0.418
0.307
0.106
0.054
-0.235
-0.187
0.426
0.357
0.094
0.053
-0.250
-0.228
et ht
w1
w2
1.814
1.540
0.634
0.288
-0.888
-0.930
2.108
1.549
0.534
0.274
-1.187
-0.944
2.150
1.809
0.476
0.268
-1.261
-1.150
Notes: The elasticities are calculated using the estimates of the two-dimensional ISLS model with
money reported in Table 1; the wage rates w1 and w2 are defined in the text.
28